# Introduction

• Kris L. G. Heyde

## Abstract

Approaching the atomic nucleus at low excitation energy (excitation energy less than the nucleon separation energy) can be done on a non-relativistic level. If we start from an A-nucleon problem interacting via a given two-body potential V i, j, the non-relativistic Hamiltonian can be written as
$$H = \sum\limits_{i = 1}^A {t_2 + \frac{1}{2}\sum\limits_{i,j = 1}^A {V_{i,j} } } ,$$
where t i is the kinetic energy of the nucleon motion. Much experimental evidence for an average, single-particle independent motion of nucleons exists, a point of view that is not immediately obvious from the above Hamiltonian. This idea acts as a guide making a separation of the Hamiltonian into A one-body Hamiltonians (described by an average one-body potential U i) and residual interactions. This can be formally done by writing
$$H = {{H}_{0}} + {{H}_{{res}}}$$
, with
$${{H}_{0}} = \sum\limits_{{i = 1}}^{A} {\left\{ {{{t}_{i}} + {{U}_{i}}} \right\}}$$
, and
$${{H}_{{res}}} = \tfrac{1}{2}\sum\limits_{{i,j = 1}}^{A} {{{V}_{{i,j}}}} - \sum\limits_{{i = 1}}^{A} {{{U}_{i}}}$$
. It is a task to determine U i as well as possible such that the residual interaction H res remains as a small perturbation on the independent A-nucleon system. This task can be accomplished by modern Hartree-Fock methods where the residual interaction with which one starts is somewhat more complicated such as the Skyrme-type interactions (two-body plus three-body terms) which have been used with considerable success. This process of going from the two-body interaction V i, j towards a one-body potential is drawn schematically.

Resid